CSE 1400 Applied Discrete Mathematics Definitions Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Arithmetic 1 Alphabets, Strings, Languages, & Words 2 Number Systems 3 Machine Numbers 3 Boolean Logic 4 Relations 5 Functions 6 Predicate (First-Order) Logic 6 Abstract Arithmetic Definition 1 (Zero). 0 is a natural number. Definition 2 (Successor Function). If n is a natural number, then σ(n) = n + 1 is a natural number called the successor of n. 0 is called a constant. 0 can be thought of as a function z(x) that returns the value 0 for all input values x. Definition 3 (Addition). If n and m are natural numbers, then n + m is a natural number called the sum of n and m. Definition 4 (Multiplication). If n and m are natural numbers, then nm is a natural number called the product of n and m. Definition 5 (Exponentiation). If n and m are natural numbers, then n m is a natural number called n raised to the exponent m. Definition 6 (Positive Exponents). n 0 = 1 n m = n n m 1
definitions 2 Definition 7 (Negation of Exponents). n m = 1 n m Definition 8 (Fractional Exponents). n a/b = b n a Definition 9 (Multipling Powers Add Exponents). n a n b = n a+b Definition 10 (Raising Powers to Powers Multiply Exponents). Definition 11 (Logarithms). (n a ) b = n ab log n a = m where n m = a Alphabets, Strings, Languages, & Words Definition 12 (Alphabet). An alphabet Σ is a finite set of characters Example 1 (Example Alphabets). The binary alphabet B = {0, 1} = {False, True} The decimal alphabet D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} The hexadecimal alphabet H = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F} The integers mod n Z n = {0, 1, 2,..., (n 1)} The quantifier alphabet (for all and there exists) Q = {, } The delimiter alphabet L = {(, ), [, ], {, }} The dna alphabet DNA = {A, C, G, T}
definitions 3 The English alphabet A = {a, b, c,..., x, y, z} The Greek alphabet G = {α, β, γ, δ, ɛ, ζ, η, θ, ι, κ, λ, µ, ν, ξ, o, π, ρ, σ, τ, υ, φ, χ, ψ, ω} The Unicode character set U = {c : 0 c (10FFFF) 16 } Definition 13 (String). A string is a finite list of characters from an alphabet Σ. The string with no characters is the empty string λ. Definition 14 (Kleene Closure). The Kleene closure Σ is the set of all strings over an alphabet Σ. Definition 15 (Language). A language L is a subset of Σ. Definition 16 (Regular Language). The empty set is a regular language. The set {λ} containing only the empty string is a regular language. For every character a Σ, the set {a} is a regular language. If L 0 and L 1 are regular languages, then the following are regular languages. The union L 0 L 1 = {s : s L 0 L 1 } The concatenation L 0 L 1 = {st : s L 0 and t L 1 } Number Systems Definition 17 (Natural Numbers). The natural numbers are the values in the set N = {0, 1, 2, 3, 4, 5,...} Definition 18 (Integers). The integers are the values in the set Z = {0, ±1, ±2, ±3, ±4, ±5,...} Definition 19 (Rationals). The rationals are the values in the set { a } Q = b : a, b Z, b = 0 The natural numbers values shown are written in decimal notation. The integer values shown are written in signed-magnitude decimal notation.
definitions 4 Machine Numbers Definition 20 (Computer Word). A fixed-length string of bits, often written in hexadecimal notation. Example 2. 8-Bit Word (00) 16 to (FF) 16 Common computer architectures have used word lengths that are multiples of 8. A string of 8 bits is called a byte. 16-Bit Word (0000) 16 to (FFFF) 16 32-Bit Word (0000 0000) 16 to (FFFF FFFF) 16 64-Bit Word Definition 21 (The Machine Naturals). The machine naturals are the values in the set N mach = {0, 1, 2, 3,..., 2 n 1} where n is the computer s word length. Definition 22 (The Machine Integers). The machine integers are the values in the set { } Z n = 2 n 1, 2 n 1 + 1,..., 1, 0, 1,..., 2 n 1 1 where n is the computer s word length. Definition 23 (The Machine Rationals (Floating Point)). The (normalized) floating point numbers are the values in the set { } Q n = ( 1) s. f 2 e b : se f = 0 is a computer word {0} The meaning of a computer word depends on context or state, often called type. n bits can name 2 n things: 0 through 2 n 1. When n is a multiple of 4, n/4 hexadecimal digits can name 16 n/4 things. Common computer architectures represent integers in two s complement notation. If m is an integer written in two s complement notation, then { (m)2 if the left-most bit of m is 0. m = m 2 n if the left-most bit of m is 1. where n is the computer s word length. Common computer architectures represent the integer exponent e in biased notation. If m is an integer written in biased notation, then where b > 0 is bias. m = (m) 2 b where s is a single bit, called the sign. e is string of bits representing an integer written in biased notation. f is string of bits representing the fractional part of the number. Boolean Logic Definition 24 (Proposition). A proposition is a statement that is either always True or always False. Definition 25 (Boolean Variable). A Boolean variable is a character, for example p, that stands for a proposition.
definitions 5 Definition 26 (Negation). p is the negation of Boolean variable p and defined by the formula False if p = True p = True if p = False Definition 27 (Negation). p is the negation of Boolean variable p and defined by the formula False if p = True p = True if p = False Definition 28 (Boolean Expressions). True and False are Boolean expressions A Boolean variable is a Boolean expressions expressions formed by applying the above operations are Boolean expressions. Definition 29 (Truth Assignment). Let Π = {p 0, p 1,..., p n 1 } be a set of Boolean variables. A truth assignment for Π is a Definition 30 (Tautology). A tautology is a Boolean expression that is always True. Definition 31 (Contradiction). A contradiction is a Boolean expression that is always False. Definition 32 (Contingency). A contingency is a Boolean expression that is sometimes True and sometimes False. And (Conjunction) p q is true only when both p and q are true. Or (Disjunction) p q is false only when both p and q are false. Conditional p q is true when (1) p is false and when (2) both p and q are true. Equivalence p q is true when both p and q are true or when both p and q are false. Relations Definition 33 (Relation). A relation r from X to Y is a subset of ordered pairs from the Cartesian product X Y. Definition 34 (Reflexive Relation). A relation r from X to X is reflexive if (x, x) r for all x X.
definitions 6 Definition 35 (Symmetric Relation). A relation r from X to X is symmetric if (x, y) r implies (y x) r. Definition 36 (Transitive Relation). A relation r from X to X is transitive if (x, y) r and (y, z) r implies (x, z) r. Definition 37 (Antisymmetric Relation). A relation r from X to X is antisymmetric if (x, y) r and (y, x) r implies x = y. Definition 38 (Equivalence Relation). An relation is an equivalence if it is reflexive, symmetric, and transitive Definition 39 (Partition). The subset S 0, S 0,..., S n 1 partition a set U if U = n 1 k=0 S i S k = S k for i = j Definition 40 (Partial Orders). A relation is a partial order if it is reflexive, antisymmetric, and transitive. Functions Definition 41 (Function). A function f from X to Y is a relation such that (x, y 1 ) f and (x, y 2 ) f implies y 1 = y 2. Definition 42 (Onto Function). A function f from X to Y is onto if for every y Y there is an x X such that (x, y 1 ) f. Definition 43 (One-to-One Function). A function f from X to Y is one-to-one if (x 1, y) f and (x 2, y) f implies x 1 = x 2. Predicate (First-Order) Logic First-order logic is a syntax capable of expressing detailed mathematical statements, semantics that identify a sentence with its intended mathematical application, and a generic proof system that is surprisingly comprehensive. Christos H. Papadimitriou, Computational Complexity Definition 44 (Predicates). A predicate is a statement that is true or false depending of the value or one or more (domain) variables. Definition 45 (Universal Quantification). A predicate can be prefixed with a universal quantifier meaning the predicate is true (or false) for all values of the domain variable(s).
definitions 7 Definition 46 (Existential Quantification). A predicate can be prefixed with an existential quantifier meaning the predicate is true (or false) for at least one value of the domain variable(s). Definition 47 (Vocabulary). A vocabulary V is a set of functions F, a set of relations R, and variable names {x, y, z, w,...} to which the functions and relations are applied. Definition 48 (Terms over a Vocabulary). 1. Any variable is a term. 2. If f is a function and t is a term, then f (t) is a term. Definition 49 (First-Order Expressions). 1. If is a relation and t and s are terms, then t s is an (atomic first-order) expression. 2. If p and q are (first-order) expressions, then the following are (first-order) expressions. (a) p, pronouced not p. (b) p q, pronouced p and q. (c) p q, pronouced p or q. (d) ( x)(p), pronouced for all x, p. Definition 50 (Models). A mo (graph theory, set theory, number theory) The negative of for all x, p is ( x)(p) = ( x)( p), pronouced there exists an x such that not p. Definition 51 (Valid Statements). Valid statements: statements that are True in every model. Definition 52 (True Statements). True statements: statements that True are of a given model. Definition 53 (Proof). Definition 54 (Provable Statements). Proveable statements: statements that can be proven True of a given model. Definition 55 (Complete). A model is complete if for every expression p there is a proof of p or there is a proof of p. Definition 56 (Inconsistent). A model is inconsistent if it there proof of every expression p. Definition 57 (Consistent). A model is consistent if it there is no there expression p such that both p and p have a proof in the theory.